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Mathematics for Physics

142 Schrodinger Equation Time-Independent

A stationary eigenvalue equation that is used to find the allowed energies and the corresponding spatial wavefunctions of a quantum system.

The time-independent Schrödinger equation. When the potential does not depend on time, a stationary state factors into a spatial wavefunction times a time-dependent phase. This principle is used to reduce the time-dependent equation to an eigenvalue problem.

The time-independent Schrödinger equation is

\[ \hat{H}\psi = E\psi \]

where

  • \(\hat{H}\) is the Hamiltonian operator.
  • \(\psi\) is the spatial wavefunction.
  • \(E\) is the energy of the state.

Its one-dimensional form. In one dimension the same equation is a second-order ordinary differential equation. This principle is used to solve bound states and scattering states on the line.

The one-dimensional time-independent Schrödinger equation is

\[ -\dfrac{\hbar^{2}}{2m}\dfrac{d^{2}\psi}{dx^{2}} + V\psi = E\psi \]

where

  • \(\psi\) is the spatial wavefunction.
  • \(V\) is the potential energy.
  • \(E\) is the energy.
  • \(m\) is the mass.
  • \(x\) is the position.

Energy eigenvalues. The allowed values of \(E\) are the eigenvalues of \(\hat{H}\). This principle is used to obtain the discrete spectrum of a bound system.

The stationary-state time factor. The full wavefunction of a stationary state is \(\Psi(x,t)=\psi(x)e^{-iEt/\hbar}\). This principle is used to recover a time-dependent solution whose probability density is constant.

Note: \(\psi\) is an eigenfunction of \(\hat{H}\).

142.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — time-independent Schrödinger equation \(\hat{H}\psi = E\psi\).
  2. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §2.1 — separation of variables.